BINARY BLACK HOLES IN CIRCULAR ORBITS: AN HELICAL KILLING VECTOR - - PowerPoint PPT Presentation

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BINARY BLACK HOLES IN CIRCULAR ORBITS: AN HELICAL KILLING VECTOR APPROACH

Eric Gourgoulhon Silvano Bonazzola Philippe Grandcl´ ement

• Phys. Rev. D. 65, 044020 (2002).
• Phys. Rev. D. 65, 044021 (2002).

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3+1 FORMALISM (VACUUM CASE)

Orthogonal projection of Einstein’s equations on spatial hypersurfaces = ⇒ split space-time into space AND time. ds2 = −

• N 2 − NiN i

dt2 + 2Nidtdxi + γijdxidxj

• Hamiltonian constraint : R + K2 − KijKij = 0
• Momentum constraints : DjKij − DiK = 0
• Evolution equations :

∂Kij ∂t − L

NKij = −DiDjN + N

• Rij − 2KikKk

j + KKij

• ∂γij

∂t − L

Nγij = −2NKij

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QUASI-STATIONARITY

• approximate but 4D space-time representing two black holes in exact circular
• rbits
• Valid when τorb. << τgrav.
• Rigorous definition of Ω.
• Circular orbit due to radiation.

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HELICAL KILLING VECTOR

Circular orbits = ⇒ Helical Killing vector l. Advance δt in time ⇐ ⇒ Rotation of δϕ = Ωδt. Inertial coordinates : lα = ∂ ∂t α + Ω ∂ ∂ϕ α Corotating coordinates :

• such as lα =

∂ ∂t α .

• coordinate t is ignorable.
• corotating shift βi = N i + Ω

∂ ∂ϕ i .

• functions N and γij are the same.

n l

α α α α

B v

N

Σt

x i’ = const. t

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ADDITIONAL HYPOTHESIS

Gauge choice : Maximum slicing K = 0. Conformal flatness approximation : γij = Ψ4fij.

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ELLIPTIC EQUATIONS

We solve 5 of the 10 Einstein’s equations :

• Hamiltonian constraint :

∆Ψ = −Ψ5 8 ˆ Aij ˆ Aij

• Momentum constraints :

∆βi + 1 3 ¯ Di ¯ Djβj = 2 ˆ Aij ¯ DjN − 6N ¯ Dj ln Ψ

• Trace of ∂Kij

∂t : ∆N = NΨ4 ˆ Aij ˆ Aij − 2 ¯ Dj ln Ψ ¯ DjN with ˆ Aij = Ψ−4Kij and ˆ Aij = Ψ4Kij. Definition of K = ⇒ ˆ Aij = 1 2N (Lβ)ij (Lβ)ij is the conformal Killing operator : (Lβ)ij = ¯ Diβj + ¯ Djβi − 2

3 ¯

Dkβkf ij Set of 5 non-linear, highly-coupled, elliptic equations.

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CHOICE OF THE TOPOLOGY

R× Misner-Lindquist.

P

a a2

1

I(P)

r1

P

a a2

1

I(P)

r2

• II

I

P I(P)

I

(t,r1 θ1 φ1 , , ) (t,r θ φ

2,

, )

2 2

R I

3

R I

3

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ISOMETRY

Mapping from MI to MII. I MI − → MII (t, xI, yI, zI) − → (t, xII = xI, yII = yI, zII = zI) (t, r1, θ1, ϕ1) − →

• t, a2

1

r1 , θ1, ϕ1

• (t, r2, θ2, ϕ2)

− →

• t, a2

2

r2 , θ2, ϕ2

• Hypothesis : the 4-metric is isometric.

∂Iµ ∂xα ∂Iν ∂xβ gµν (I (P)) = gαβ (P) Consequence : solve only on MI with boundary conditions on the throats.

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BOUNDARY CONDITIONS ON THE THROATS

N (I (P)) = ±N (P) = ⇒    N|Si = ( or ∂r N|Si = 0)          βr (I (P)) = −a2 r2 βr (P) βθ (I (P)) = βθ (P) βϕ (I (P)) = βϕ (P) = ⇒                    βr|Si = ∂θβr|Si = ∂ϕβr|Si = ∂rβθ

• Si

= ∂rβϕ|Si = Ψ (I (P)) = a r Ψ (P) = ⇒ ∂rΨ + 1 2aΨ

• Si

= 0 Consequence: Kij (I (P)) = − ∂Ii ∂xk ∂Ij ∂xl Kkl (P) The throats are apparent horizons.

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STATE OF ROTATION

• β not completely fixed by isometry.

Corotating black holes = ⇒ β

• Si

= 0. Properties :

• Analogy with rigidity theorem.
• Throats are Killing horizons.

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ISOMETRY AND REGULARITY

Rigidity implies isometry except : ∂rβθ

• Si = 0

and ∂rβϕ|Si = 0 To have a regular K : ˆ Aij = (Lβ)ij 2N N|Si =      = ⇒ (Lβ)ij

• Si = 0.

So to have RIGIDITY, REGULARITY and ISOMETRY one must have :

• β
• Si

= ∂r β

• Si

= In this framework it is impossible to have REGULARITY for non-corotating black holes.

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REGULARIZATION OF THE SHIFT

One solves for β, using Dirichlet-type boundary condition :

• β
• Si

= 0 At each iteration one modifies the shift vector by :

• βnew =

β + βcor

• βcor is chosen so that :
• βnew
• Si

= ∂r βnew

• Si

= 0. At the end of a calculation :

• if

βcor → 0 : exact solution.

• if

βcor is small : approximate solution.

• else not a solution !

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BOUNDARY CONDITIONS AT INFINITY

To recover Minkowski space-time : N → 1 when r → ∞ Ψ → 1 when r → ∞

• β → Ω ∂

∂ϕ when r → ∞

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DETERMINATION OF Ω

Ω only present in the boundary condition for β. One can solve for ANY value of Ω (example : Ω = 0 = ⇒ Misner-Lindquist). SUPPLEMENTARY CONDITION : the O

• r−1

part of the metric when (r → ∞) is identical to Schwarzschild. A priori : Ψ ∼ 1 + MADM 2r and N ∼ 1 − MK r One chooses the ONLY Ω such that : MK = MADM ⇐ ⇒ Ψ2N ∼ 1 + α r2 Justifications :

• exact stationary asymptotical space-times.
• Newtonian limit =

⇒ virial theorem.

• True for binary neutron stars.

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CONSTRUCTION OF A SEQUENCE

Each configuration depends on 2 parameters :

• radius of the throats a.
• separation D

a . Existence of a scaling factor : only one sequence. a is chosen so that : dMADM dJ

• sequence

= Ω.

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AREA OF THE HORIZONS

First law of thermodynamics for binary black holes : dMADM = ΩdJ + 1 8π (κ1dA1 + κ2dA2) Consequence : along the sequence dA = 0. Horizon area is constant : quasi-static evolution (Second law of thermodynamics for black holes).

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NUMERICAL METHODS

Basic features :

• Multi-domain : two sets of spherical coordinates.
• Compactification : exact treatment of spatial infinity.
• Spectral decomposition : spherical harmonics and Tchebychev polynomials.

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NORM OF βcor

10 15 20 25 30 Separation parameter D/a 10

• 4

10

• 3

10

• 2

10

• 1

Relative error Correction (21*17*16) Error on J (21*16*17) Correction (33*21*20) Error on J (33*21*20)

J∞ = JS⇐ ⇒ βcor = 0

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SMARR FORMULA

10 15 20 25 Separation parameter D/a 10

• 4

10

• 3

10

• 2

10

• 1

Relative error on Smarr formula J infinity (21*17*16) J throats (21*17*16) J infinity (33*21*20) J throats (33*21*20)

M − 2ΩJ = − 1 4π

• S1

Ψ2 ¯ DiNd ¯ Si − 1 4π

• S2

Ψ2 ¯ DiNd ¯ Si

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AREA OF THE HORIZONS

0.05 0.075 0.1 0.125 0.15 0.175 0.2 Orbital velocity ΩMir 1e-04 1e-03 Relative change of the irreducible mass Mir 21*17*16 33*21*20

Mir =

• A1

16π +

• A2

16π

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LAPSE IN THE ORBITAL PLANE

ISCO configuration

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CONFORMAL FACTOR IN THE ORBITAL PLANE

ISCO configuration

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KXY IN THE ORBITAL PLANE

ISCO configuration

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SEQUENCE : VARIATION OF MADM

0,05 0,1 0,15

Ω Mir

0,98 0,985 0,99 0,995

MADM / Mir

Grandclement et al. 2001, 33x21x20 Grandclement et al. 2001, 21x17x16 EOB 3-PN a4=4.67 corot, Damour et al. 2001 EOB 3-PN a4=4.67 irrot, Damour et al. 2000 Cook 1994, Pfeiffer et al. 2000, irrot

Comparison Numerical results <-> 3-PN EOB

Total energy along a sequence

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SEQUENCE : VARIATION OF J

0,05 0,1 0,15

Ω Mir

0,8 0,9 1

J / Mir

2

Grandclement et al. 2001, 33x21x20 Grandclement et al. 2001, 21x17x16 EOB 3-PN a4=4.67 corot, Damour et al. 2001 EOB 3-PN a4=4.67 irrot, Damour et al. 2000 Cook 1994, Pfeiffer et al. 2000, irrot

Comparison Numerical results <-> 3-PN EOB

Total angular momentum along a sequence

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POSITION OF THE ISCO

0.05 0.1 0.15 0.2 0.25 0.3

Orbital velocity Ω Mir

−0.03 −0.025 −0.02 −0.015

Binding energy (M−Mir) / Mir

33*21*20 21*17*16 3−PN, corotating, ωs=0 (Damour et al. 2001) 3PN, S=0, ωs=−9 (Damour et al. 2000) Conformal imaging S=0 (Pfeiffer et al. 2000) Conformal imaging S=0.08 (Pfeiffer et al. 2000) Conformal imaging S=0.17 (Pfeiffer et al. 2000) Puncture S=0 (Baumgarte 2000)

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PLAUSIBLE EXPLANATION

Main difference with IVP : extrinsic curvature tensor IVP HKV choice given by the shift vector Ψ2Kij = 3 2r2

• Pinj + Pjni − (fij − ninj) P knk
• Ψ4Kij =

1 2N (Lβ)ij Indications that IVP does not produce real circular orbits :

• Plunge even for pre-ISCO initial conditions =

⇒ the real ISCO is further away.

• One must impose 0.55Ω to maintain the black holes in the corotating frame

= ⇒ Ω is to big.

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SO WHAT ?

Realistic conclusions :

• The problem is not solved !
• This approach probably not complete crap !
• Conformal flatness approximation not so bad.
• Results using “standard” IVP should be questioned.

The future (in chronological order, with time-scales) :

• Use real apparent boundary conditions (few months).
• Construct sequences with other states of rotation and with different masses

(few months).

• Relax the conformal flatness approximation (1-2 years).
• Investigate the K = 0 hypothesis.